Advanced Mathematics MCQs

41

Advanced Mathematics • Complex Analysis

If a function f(z), defined on some domain D of the complex plane, is differentiable on D, it is said to be: If a function f(z), defined on some domain D of the complex plane, is differentiable on D, it is said to be:

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A. Analytic function Analytic function B. Non-singular function Non-singular function C. Bounded function Bounded function D. Harmonic function Harmonic function

सही उत्तर: A Correct Answer: A

A complex function differentiable at every point of a domain is analytic, or holomorphic, on that domain.

42

Advanced Mathematics • Complex Analysis

Let f(r cos(theta) + i r sin(theta)) = u(r,theta) + iv(r,theta). For f to be differentiable at (r,theta), the Cauchy-Riemann equations state that: Let f(r cos(theta) + i r sin(theta)) = u(r,theta) + iv(r,theta). For f to be differentiable at (r,theta), the Cauchy-Riemann equations state that:

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A. u_r = -v_r/r, v_r = -u_theta/r u_r = -v_r/r, v_r = -u_theta/r B. u_r = v_theta/r, v_r = -u_theta/r u_r = v_theta/r, v_r = -u_theta/r C. u_r = -v_theta/r, v_theta = -u_theta/r u_r = -v_theta/r, v_theta = -u_theta/r D. u_r = -v_theta/r, v_r = -u_theta/r u_r = -v_theta/r, v_r = -u_theta/r

सही उत्तर: B Correct Answer: B

In polar coordinates the Cauchy-Riemann equations are u_r = v_theta/r and v_r = -u_theta/r.

43

Advanced Mathematics • Linear Programming

A solution to a linear programming problem that satisfies all the constraints except the non-negativity constraint is referred to as: A solution to a linear programming problem that satisfies all the constraints except the non-negativity constraint is referred to as:

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A. Feasible solution Feasible solution B. Basic solution Basic solution C. Optimal solution Optimal solution D. Basic feasible solution Basic feasible solution

सही उत्तर: B Correct Answer: B

A basic solution satisfies the system of equality constraints but need not satisfy non-negativity. If it also satisfies non-negativity, it becomes a basic feasible solution.

44

Advanced Mathematics • Partial Differential Equations

The equation of the characteristic curve of the one-parameter family (x - a)^2 + (y - a)^2 + z^2 = 1 is: The equation of the characteristic curve of the one-parameter family (x - a)^2 + (y - a)^2 + z^2 = 1 is:

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A. (x - y)^2 = 2(1 - z^2) (x - y)^2 = 2(1 - z^2) B. (x - y)^2 = 1 - z^2 (x - y)^2 = 1 - z^2 C. (x - z)^2 = 2(1 - y^2) (x - z)^2 = 2(1 - y^2) D. (x - z)^2 = 1 - y^2 (x - z)^2 = 1 - y^2

सही उत्तर: A Correct Answer: A

Differentiating with respect to the parameter gives a = (x + y)/2. Eliminating a from the family yields (x - y)^2 = 2(1 - z^2).

45

Advanced Mathematics • Complex Analysis

If C is a positively oriented simple closed contour within and on which f is analytic, except for a finite number of singular points z_k (k = 1,2,...,n) interior to C, then the Residue Theorem states that: If C is a positively oriented simple closed contour within and on which f is analytic, except for a finite number of singular points z_k (k = 1,2,...,n) interior to C, then the Residue Theorem states that:

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A. Integral_C f(z)dz = e pi i (sum_k Res f(z_k)/k!) Integral_C f(z)dz = e pi i (sum_k Res f(z_k)/k!) B. Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)) Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)) C. Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)/k!) Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)/k!) D. Integral_C f(z)dz = sum_k Res f(z_k) Integral_C f(z)dz = sum_k Res f(z_k)

सही उत्तर: B Correct Answer: B

By the Residue Theorem, the contour integral equals 2pi i times the sum of the residues at all singularities inside C.

46

Advanced Mathematics • Probability

Consider 10 tosses of a biased coin with probability of obtaining a head equal to 1/3. Let Y denote the total number of tails observed. Find E(5Y + 2). Consider 10 tosses of a biased coin with probability of obtaining a head equal to 1/3. Let Y denote the total number of tails observed. Find E(5Y + 2).

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A. 10 10 B. 56/3 56/3 C. 106/3 106/3 D. 10/3 10/3

सही उत्तर: C Correct Answer: C

The probability of a tail is 2/3, so E(Y) = 10 x 2/3 = 20/3. Thus E(5Y + 2) = 5E(Y) + 2 = 106/3.

47

Advanced Mathematics • Graph Theory

A graph G with p vertices (p >= 3) is Hamiltonian if: A graph G with p vertices (p >= 3) is Hamiltonian if:

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A. For all vertices v in G, degree(v) < p/2 For all vertices v in G, degree(v) < p/2 B. For all vertices v in G, degree(v) >= p/2 For all vertices v in G, degree(v) >= p/2 C. All vertices v in G have an even degree All vertices v in G have an even degree D. All vertices in G have an odd degree All vertices in G have an odd degree

सही उत्तर: B Correct Answer: B

By Dirac's theorem, a simple graph with p >= 3 vertices is Hamiltonian if every vertex has degree at least p/2.

48

Advanced Mathematics • Statics

If P and Q are two non-intersecting forces whose directions are perpendicular, then the ratio of the distances of the central axis from their lines of action is represented as: If P and Q are two non-intersecting forces whose directions are perpendicular, then the ratio of the distances of the central axis from their lines of action is represented as:

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A. Q^2 : 2P^2 Q^2 : 2P^2 B. 2P^2 : Q^2 2P^2 : Q^2 C. Q^2 : P^2 Q^2 : P^2 D. P^2 : Q^2 P^2 : Q^2

सही उत्तर: C Correct Answer: C

For two perpendicular non-intersecting forces, the distances of the central axis from their respective lines of action are in the inverse-square ratio of the forces, giving Q^2 : P^2 in the stated order.

49

Advanced Mathematics • Complex Analysis

Let C denote the boundary of the circle with radius 3 around the centre z_1 = 2 + i. Calculate the integral I = integral_C e^(az)/(z - 2 - i) dz. Let C denote the boundary of the circle with radius 3 around the centre z_1 = 2 + i. Calculate the integral I = integral_C e^(az)/(z - 2 - i) dz.

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A. 0 0 B. 2pi i e^(a(2 - i)) 2pi i e^(a(2 - i)) C. 2pi i e^(a(2 + i)) 2pi i e^(a(2 + i)) D. 2pi i e^(a(2 + 2i)) 2pi i e^(a(2 + 2i))

सही उत्तर: C Correct Answer: C

By Cauchy's integral formula, the integral equals 2pi i times e^(az) evaluated at z = 2 + i, giving 2pi i e^(a(2 + i)).

50

Advanced Mathematics • Probability

The density of a random variable X is f_X(x) = a + bx^2 for x in [0,1], and 0 otherwise. Find (a,b) if E(X) = 3/5. The density of a random variable X is f_X(x) = a + bx^2 for x in [0,1], and 0 otherwise. Find (a,b) if E(X) = 3/5.

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A. (3/5, -3/5) (3/5, -3/5) B. (3, -3) (3, -3) C. (1, 2) (1, 2) D. (3/5, 6/5) (3/5, 6/5)

सही उत्तर: D Correct Answer: D

Normalization gives a + b/3 = 1, while E(X) = a/2 + b/4 = 3/5. Solving yields a = 3/5 and b = 6/5.

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